Abstract
A human being standing upright with his feet as the pivot is the most popular example of the stabilized inverted pendulum. Achieving stability of the inverted pendulum has become common challenge for engineers. In this paper, we consider an initial value discrete fractional Duffing equation with forcing term. We establish the existence, Hyers–Ulam stability, and Hyers–Ulam Mittag-Leffler stability of solutions for the equation. We consider the inverted pendulum modeled by Duffing equation as an example. The values are tabulated and simulated to show the consistency with theoretical findings.
Highlights
The understanding of the real-world problems by replicating into mathematical models proves to be an effective tool
A model of heart beat was constructed using an electrical circuit with coupled relaxation oscillators and simulations of normal heart beat and of certain disorders were convincingly obtained by Van der Pol and Van der Mark [3]
Certain damped and driven oscillators are modeled by the Duffing equation, a second-order differential equation with cubic nonlinearity named after Georg Duffing [4]
Summary
The understanding of the real-world problems by replicating into mathematical models proves to be an effective tool. Qualitative analysis of the solutions of fractional-order equations representing real-life phenomena plays a predominant role in understanding the nature and behavior of the models [7, 8]. Chen [23, 24] was the first author who studied the stability results of the nonlinear fractional difference equations. We consider the discrete-time forced fractional-order Duffing equation without damping.
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